| Title:
|
On interval homogeneous orthomodular lattices (English) |
| Author:
|
de Simone, A. |
| Author:
|
Navara, M. |
| Author:
|
Pták, P. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
42 |
| Issue:
|
1 |
| Year:
|
2001 |
| Pages:
|
23-30 |
| . |
| Category:
|
math |
| . |
| Summary:
|
An orthomodular lattice $L$ is said to be interval homogeneous (resp. centrally interval homogeneous) if it is $\sigma$-complete and satisfies the following property: Whenever $L$ is isomorphic to an interval, $[a,b]$, in $L$ then $L$ is isomorphic to each interval $[c,d]$ with $c\leq a$ and $d\geq b$ (resp. the same condition as above only under the assumption that all elements $a$, $b$, $c$, $d$ are central in $L$). Let us denote by Inthom (resp. Inthom$_c$) the class of all interval homogeneous orthomodular lattices (resp. centrally interval homogeneous orthomodular lattices). We first show that the class Inthom is considerably large — it contains any Boolean $\sigma$-algebra, any block-finite $\sigma$-complete orthomodular lattice, any Hilbert space projection lattice and several other examples. Then we prove that $L$ belongs to Inthom exactly when the Cantor-Bernstein-Tarski theorem holds in $L$. This makes it desirable to know whether there exist $\sigma$-complete orthomodular lattices which do not belong to Inthom. Such examples indeed exist as we than establish. At the end we consider the class Inthom$_c$. We find that each $\sigma$-complete orthomodular lattice belongs to Inthom$_c$, establishing an orthomodular version of Cantor-Bernstein-Tarski theorem. With the help of this result, we settle the Tarski cube problem for the $\sigma$-complete orthomodular lattices. (English) |
| Keyword:
|
interval in a $\sigma$-complete orthomodular lattice |
| Keyword:
|
center |
| Keyword:
|
Boolean $\sigma$-algebra |
| Keyword:
|
Cantor-Bernstein-Tarski theorem |
| MSC:
|
06C15 |
| MSC:
|
06E05 |
| MSC:
|
81P10 |
| idZBL:
|
Zbl 1077.06005 |
| idMR:
|
MR1825370 |
| . |
| Date available:
|
2009-01-08T19:08:14Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119221 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| . |
